Network Fact @NetworkFact
Daily tweets about networks, graph theory, and related topics from @JohnDCook JohnDCook.com/networks Joined July 2013-
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If A is the adjacency matrix of a graph, the trace of A³ is six times the number of triangles in the graph.
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Visualizing Treewidth arxiv.org/abs/2508.19935
The characteristic polynomial of a graph is the determinant of xI - A where A is its adjacency matrix.
Given a list of numbers, when can you construct a simple graph whose nodes have those degrees? en.wikipedia.org/wiki/Erd%C5%91…
Modularity in a network is a measure of the degree to which nodes associate with like nodes. en.wikipedia.org/wiki/Modularit…
Graphing Japanese prefectures johndcook.com/blog/2022/09/1…
The eigenvector corresponding to the second smallest eigenvalue of the Laplacian is called the Fiedler vector.
Hitting time is asymmetric. Commute time c(i, j) = h(i -> j) + h(j -> i) is symmetric.
The hitting time h(i -> j) is the average number of steps to walk from node i to node j where probability is proportional to edge weight.
Why are planar graphs so exceptional? mathoverflow.net/questions/7114…
The reciprocity of a graph is the proportion of edges that are reciprocated.
The betweenness centrality of a point is the number of geodesic paths that pass through the point.
The Petersen graph serves as a counterexample to many would-be graph conjectures. en.wikipedia.org/wiki/Petersen_…
Basic theory of expander graphs by Terence Tao terrytao.wordpress.com/2011/12/02/245…
A knight's random walk on a graph johndcook.com/blog/2012/05/1…
A regular graph is one in which all nodes have the same degree.
Laboratory for Web Algorithmics law.di.unimi.it/index.php
Applied algebraic topology and sensor networks math.upenn.edu/~ghrist/prepri… [24 MB pdf]
Python software for compressed sparse graphs: docs.scipy.org/doc/scipy/refe…
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